Measuring interest rates as determined bythrift and productivity

January 28, 2000

Woon Gyu Choi*

Department of Economics, Hong Kong University of Science & TechnologyClear Water Bay, Kowloon, Hong Kong

E-mail: wgchoi@usthk.ust.hk

and

Yi Wen*

Department of Economics, Cornell UniversityIthaca, NY 14853, USA

E-mail: yw57@cornell.edu

ABSTRACT

This paper investigates the behavior of short-term real and nominal rates of interest bycombining consumption-based and production-based models into a single general equilibriumframework. Based on the theoretical nonlinear relationships that link interest rates to both themarginal rates of substitution and transformation in a monetary production economy, wedevelop an estimation and simulation procedure to generate historical time series of interestrates. We find that the predictions of interest rates based on a general equilibrium theory arepartially consistent with US data.

Keywords: Interest rate; Marginal rate of substitution; Marginal rate of transformation; General equilibrium; Assetpricing

JEL Classification: C22 Time series models ; E43 Determination of interest rate

________________________________* We are grateful to Axel Leijonhufvud, Simon Potter, Gary Hansen, Roger Farmer, Patrick Asea, Keunkwan Ryu,and Jeff Armstrong for useful suggestions on an earlier version of this paper.

1

1. Introduction

This paper attempts to explain the dynamic behavior of interest rates in the US economy using

an equilibrium theory. We investigate the behavior of interest rates from two directions of the

economy: thrift and productivity. The first direction comes from a relationship between interest

rates and (intertemporal) marginal rates of substitution (MRS). It is thrift that links the supply of

funds to interest rates. The second one comes from a link between interest rates and marginal rates

of transformation (MRT) that depends on the marginal productivity of capital. The equilibrium rate is

determined at a level that balances thrift and productivity.

These two directions, although they have been examined separately in the asset pricing

literature, have not been examined simultaneously using a single general equilibrium framework.

Most existing asset pricing models use a consumption-based approach and focus only on the MRS

in an endowment economy (see Kocherakota, 1996). There has been little attention paid to the

dynamics of interest rates from the point of view of the MRT using a production-based approach.

The important exceptions are studies by Cochraine (1991, 1996), who uses the production-based

approach. This work, however, is based on partial equilibrium models.1 The MRT aspect of interest

rates is interesting to consider because it reflects very different mechanisms of interest rate dynamics

from those implied by the MRS. By examining these two directions simultaneously in a single general

equilibrium framework,2 we may also be able to identify any inconsistency between the implications

of MRS and MRT in a general equilibrium theory. In this paper, we use a limited participation

model for a monetary production economy to examine interest rate behavior from the viewpoint of

consumers (MRS) and producers (MRT) at the same time.

In a monetary economy, interest rates depend not only on the fundamentals of thrift and

productivity but also on monetary factors, which operate together in combined forms. What we

observe is the nominal interest rate that can be decomposed into the real interest rate and the

1 Using a production (or investment)-based model, Cochrane (1991, 1996) explains the market return with aninvestment return, inferred from investment data with an adjustment cost production function. The investmentreturn is constructed from a regression analysis, i.e., regressing the investment return on its motivating factorsimplied by the producer’s first-order conditions.2 Den Haan (1995) considers a simple production economy that can generate persistence in the interest rate andthe slope of the term structure. However, he looks at only one direction of the production economy and does notexamine the historical movement of interest rates.

2

expected inflation. By linking up with consumption side and production side of the economy, we

explain the behavior of interest rates by the fundamentals that are entangled with the inflation

process. The nominal interest rates will eventually rise by the same amount as any increase in fully

anticipated inflation through the so-called Fisher effect. The natural rate of interest reflects the

equality between saving and investment, being determined by both thrift and productivity. The

market interest rate is affected by the excess demand for the loanable funds that are provided by the

banking system, as implied by the liquidity effect.3

This paper sheds light on the question of whether or not the general equilibrium framework

utilized extensively in the business cycle literature can predict the historical movement of interest

rates. It is well known that interest rates are leading indicators of business cycles (e.g., Bernanke

and Blinder, 1992). However, most existing studies that attempt to explain the behavior of interest

rates based on a general equilibrium framework have focused on the mean and variance (e.g., Weil,

1989; Cecchetti et al., 1993; Abel, 1994; Den Haan, 1995) or covariation with inflation (Giovannini

and Labadie, 1991), not on the dynamic properties of interest rates, which would have important

implications for investment and output fluctuations. This paper also provides some insights on the

risk-free rate puzzle by looking at the production side of the economy.

We start our investigation by deriving the exact theoretical relationships that link interest rates to

the MRS on the one hand and to the MRT on the other, using a monetary general equilibrium model

featuring the liquidity effect (e.g., Lucas, 1990; Christiano, 1991; Fuerst, 1992; Dow, 1995). We

then calibrate the nominal and real interest rates using the actual time series on the fundamentals

(such as the consumption growth, inflation, and marginal product of capital) to investigate whether

the calibrated MRS and MRT explain the historical movements of the rate of return on the three-

month Treasury bill.4 Since these relationships are highly nonlinear and they involve conditional

expectations, we use a simulation method to calibrate theoretical interest rates.

3 Saving equals the supply of loanable funds by households and investment equals the demand for loanablefunds by firms. If the banking system just intermediates household saving without generating any net injectionof loanable funds on its own, the economy will adjust so that the interest rate is driven to its natural level.4 Although MRS = MRT in equilibrium, they may nevertheless give very different predictions of calibratedinterest rates as they embody different sets of fundamentals and thus contain different information sets aboutthe actual economy , in addition to Euler equation errors.

3

We examine the consistency between the model prediction and actual data in terms of

autocorrelation and spectral density functions and further investigate whether our model can predict

the historical movements of the data. We find that, for the real rate, MRT explains the dynamics of

the data remarkably well and outperforms MRS. We also find that, for the nominal rate, both MRT

and MRS have explanatory power at the business cycle frequency but not at higher frequencies.

The results suggest that our attempt to explain the interest rate behavior by jointly examining

consumers and producers’ viewpoints is worthwhile and that the dynamic properties of interest rates

can be explained to a substantial extent by the fundamentals of thrift and productivity. Nonetheless,

there still remains a long way to go for the general equilibrium theory to explain the short-term (high

frequency) dynamics of nominal interest rates.

The remainder of this paper is organized as follows. Section 2 describes the relationships linking

interest rates to MRS and MRT and derives the stochastic process of endogenous variables from a

general equilibrium model. Section 3 develops estimation and simulation methods for calibrating

interest rates series. Section 4 presents our empirical results with discussion after a brief description

of the data construction. Section 5 concludes.

2. Explaining Interest Rates as Determined by Thrift and Productivity

We use a standard monetary general equilibrium model based on Lucas (1990), Christiano

(1991), Fuerst (1992), and Dow (1995). We assume a closed economy since we examine the

interest rate behavior in the US economy, which is large enough to abstract influences from abroad.

In a growing monetary production economy with portfolio rigidity and capital accumulation, cash-in-

advance constraints (CIA) are imposed on all types of transactions, and households choose

portfolios before the current state is known. The economy is lumped into families consisting of

multiple members, one of which is the financial intermediary that creates the newly injected cash in

4

the financial market.5 The model is driven by monetary shocks that call for the liquidity and Fisher

effects on interest rates.6

2.1. The Model

The population consists of many identical families. The number of members of a family, nt,

grows with growth factor η, while the number of families is fixed. The representative family

(household) has preferences over uncertain consumption and leisure streams given by

,)1ln()(0

0

−+∑

∞

=ttt

tt lcUE ηβ 0>β , (2.1)

where E0 is the expectations operator conditional on period 0, β is the subjective discount factor, ct

is per capita consumption, tl is work effort (leisure endowment is normalized to one), and

)1/()1()( 1 γγ −−= −tt ccU with the relative risk aversion coefficient of γ.

A multiple-member household consists of a shopper, a worker, a firm and a financial

intermediary (bank). First, each firm has access to a production technology. With a constant

depreciation rate )1,0(∈δ , the law of motion for the per capita capital stock is

ttt ikk +−=+ )1( 1 δη , (2.2)

where it is investment. Let the labor in efficiency units grow at the rate υ − 1. Then the effective

labor unit in period t is given by ttE

t hh υ= , where ht is the labor input. The firm produces output,

qt, by means of a Cobb-Douglas function:7

(0,1) ,),( 1)1( ∈== −− αυ αααtt

tttt hkhkfq .

5 As in Lucas (1990), the effect of monetary injections are symmetric across families, but asymmetric within thefamily since only firms are forced to absorb monetary shocks introduced via the financial intermediary. For theliterature on the non-neutrality generated from the asymmetry, see Fuerst (1992) and Christiano (1991).6 The money supply change has the first-round loanable funds effect through changes in the excess supply ofloanable funds (Friedman and Schwartz, 1982, Ch 10). The liquidity effect represents that an exogenous increasein the money supply forces down the interest rate through agents’ portfolio adjustments. Since these two effectscannot be separated from each other in the actual data, we shall refer to them simply as the liquidity effect.7 Our model has a balanced-growth equilibrium. The log of output has a trend component and preferences arerestricted so that technological progress has no long–run effect on labor supply. For specifications of balanced-growth models, see King et al. (1988) and Cooley and Prescott (1995). This paper, for simplicity, abstracts fromproductivity shocks. The incorporation of stochastic productivity shocks does not affect our main results.

5

Second, each household consists of a worker/shopper pair: workers sell their labor services to

firms, and shoppers purchase goods from firms. Third, money is introduced via CIA constraints on

all transactions. Fourth, a household starting with Jt dollars chooses to deposit Dt of these balances

in the financial intermediary before the state of the world for period t is revealed. After this deposit is

made, the family separates and the worker travels to the labor market, while both the intermediary

and firm travel to the credit market. Fifth, after separation, the state determines the monetary

injection (X t ) that is given to each financial intermediary. The representative financial intermediary

has D Xt t+ dollars to lend out. The firm borrows Bt at the nominal interest rate Rt from the

intermediary. The firm then hires workers at the wage Wt and sells the current product in the goods

market at the price Pt .

Furthermore, we assume that the firm sells ( )1− δ kt existing net capital and purchases 1+tkη

future capital stock by the end of period t (i.e., selling and repurchasing the existing capital stock).

The firm borrows from the bank the amount to purchase investment goods ( tt kk )1(1 δη −−+ ) and to

pay wages. The firm pays the rental cost for the existing capital stock during the period of producing

and selling of its product.

Let Mt be the economy-wide per capita money stock. The economy-wide per capita money

stock, Mt , follows the law of motion given by

η M M xt t t+ = +1 1( ) , (2.3)

where xt , the per capita money stock growth, follows a stochastic process that will be specified

later. We measure all nominal variables in period t relative to the start-of-period aggregate money

stock per family, n Mt t . Then m J n Mt t t t= / ( ) denotes a household's per capita money holdings

relative to the economy-wide per capita money holdings. Similarly, define d D n Mt t t t= / ( ),

b B n Mt t t t= / ( ) , ttt PWw /= , pt = P n Mt t t/ ( ) , and x X n Mt t t t= / ( ) .

Let V m k( , , )κ denote the maximized objective function for the representative household that

begins a period with m cash balance, k capital stock, and κ, economy-wide per capita capital

stock. Now the dynamic optimization motivates the Bellman equation:

+−+= +++−

+

),,( )1()(maxmax),,( 111,,,,

1 1

ttttttklbhc

td

ttt kmVlWcUEEkmVtttttt

κβηκ (2.4)

6

subject to:,tttt cpdm ≥− (2.5)

,))1(( 1 tttttt hwkkpb +−−≥ + δη (2.6)

.)1(]))1((),([ )1(

1

1

ttttttttttt

tttttttttt

xRbkkphwhkfp

RxcplwRdmm

+−−−−−+++−++=

+

+

δη (2.7)

CIA constraints (2.5)−(2.6) apply to the shopper and to the firm, respectively. The law of motion

for money balance is given by Eq. (2.7). The cash balance that a family will have at the start of

period t+1 is contributed by the worker/shopper ( ttttttt cphwRdm −++ ), the intermediary

( )1( tt Rx + ), and the firm [ ttttt hwhkfp −),( ttttt Rbkkp −−−− + ))1(( 1 δη ].

2.2. Marginal Rate of Substitution and Marginal Rate of Transformation

On the supply side of loans, the interest rate is linked to the (intertemporal) marginal rates of

substitution (MRS) in consumption between periods t and t+1. From the equilibrium conditions, we

have the relationship that links interest rates to the MRS (see Appendix A):

,]/)(' []/)('[1 11 +++Λ=+ ttttttt PcUEPcUR βη (2.8)

where Λt t t t tn M= −( ) / ( )λ λ2 1 measuring the liquidity effect, λ1 is the liquidity cost incurred by

not holding cash, and λ2 summarizes the borrowing cost of the firm (λ1 and λ2 denote the Lagrange

multipliers associated with Eqs. (2.5) and (2.6), respectively). The banks decide the bank rate and

provide signals to the differentiated groups of the economy via the interest rate. Without an accurate

forecast, the value of money will not be equally valued in the goods and financial markets, i.e.,

λ λ1 2t t≠ . Thus, the liquidity effect arises from monetary injections through the banks and the

forecast errors attributed to an information structure wherein households make portfolio choices

before all state information is available.

One can rewrite Eq. (2.8) as the Fisherian decomposition of the nominal rate into the real rate

and expected inflation treating the unobservable liquidity effect as a stochastic error:

tt

ttttt P

PcUEcUR ξβη +

⋅=+

++ )(' )('1

11 , (2.9)

where ξt is a stochastic error with zero mean due to the liquidity effect.

7

On the other hand, the demand for loans depends on the evaluated return from investment in

physical capital and the opportunity cost of borrowing in the financial market. From the equilibrium

conditions, we obtain the following relationship (see Appendix A):

)1/()()()1/()()1( 1211,1 +++++ +−+=++ ttmttktttmtt xkVpfExkVpR βηδη , (2.10)

where /)(')( 111 +++ = ttttm pcUEkV . This represents the condition that the evaluated return from

current investment equals the cost of the borrowed cash balance. Suppose a firm borrows Pt dollars

from the bank at the cost of interest Rt per dollar. If the firm uses borrowed money for production,

it will have, at time t+1, the expected cash flow of ])[( 11, ++ −+ ttkt PfE δη that can be used for

future purposes with the per dollar gain summarized by )1/()( 12 ++ + ttm xkVβη . Note that it is not

until the start of period t+2 that the firm can use this return for future purposes.8 Then equilibrium

conditions require that the expected value of the marginal productivity of capital net of depreciation

that can be used for future purposes at the start of period t+2 should be equal to the value of

borrowed funds that can be used for future purposes at the start of period t+1.

Using p P n Mt t t t= / ( ) , n nt t+ =1 η , and Eq. (2.3), we can rewrite Eq. (2.10) to obtain the

relation that relates the interest rate to the marginal rate of transformation (MRT):

−+=+ ++

++

++ )(')(')(1 1

12

2

11, t

t

ttt

t

ttktt cU

PP

EcUPP

fER δηβη . (2.11)

Returns from alternative uses of cash relate the interest rate to the thrift factor (MRS) and

productivity factor (MRT). The equilibrium rate will be determined at the level that satisfies the

condition MRS=MRT, which corresponds to the Keynes-Ramsey rule under certainty.9

3. Methodology of Generating Interest Rate Series

8 For the return on the investment to be liquidated, two periods are required: one for the gestation period ofcapital and the other for selling the product and distributing the return.9 The rate of return on deposit (Rt) is taken to be the risk-free, nominal interest rate. The introduction of a nominaldiscount bond in zero net supply does not affect conditions (2.9) and (2.11), and renders Rt equal to the rate ofreturn on the (default-free) bond. Hence Rt is considered to be the market interest rate.

8

According to Eqs. (2.9) and (2.11), the nominal interest rate can be explained in two ways. One

pertains to the consumption-based approach that links the interest rate to the thrift factor. Another

pertains to the production-based approach that links the interest rate to the productivity factor. In

this section, we test whether these approaches are consistent with US data. Namely, we calibrate

interest rates according to Eqs. (2.9) and (2.11) using the actual data on the consumption growth,

inflation, and marginal product of capital, and compare the calibrated series with the actual series.

The (ex-ante) real rate is then determined by deflating the nominal rate by the expected inflation

factor.

Although the theoretical relationships involve nonlinear expectations, the conditional normality of

the consumption growth, inflation, and output-capital ratio in logarithm, consistent with our

theoretical model, helps reduce them to linear expectations involving only the conditional mean and

covariance of the observables.10 We then use a vector autoregression (VAR) model to estimate the

conditional mean and covariance of the observables. To ensure full consistency between theory and

estimation, we impose the VAR structure implied by our theoretical model on the data in estimation.

Namely, we estimate the constrained vector autoregressive moving average (CVARMA) model that

is implied by the autoregressive moving average (ARMA) structure of the control variables. The

CVARMA generates forecasts for the consumption growth, inflation, and capital-output ratio, from

which we compute conditional expectations and a covariance matrix of forecast errors for relevant

variables. Finally, predictions are made about interest rates by simulating the theoretical relationships

after replacing conditional expectations and covariance with their estimates.

3.1. ARMA Structure of the Theoretical Model

To derive the stochastic processes of variables on which the conditional expectations are based,

consider the log-linearized equilibrium conditions of the model. The information set at the beginning

of period t includes the stock of capital, k t, and the exogenous money growth shock, x t. Since the

decision on the deposit, dt, has to be made before the shock is realized, it is a function of k t and x t-1,

10 This approach to calibrate a time series is in line with previous studies such as Cochrane (1991) and Watson(1993). Assuming the log-normality of the driving shocks of the economy, the Euler equations are approximatedas log-linear functions for endogenous variables (e.g., Campbell 1994), which are expressed as a CVARMA modelin this paper.

9

abstracting from a constant term for simplicity. As a result, the state space of the model contains

innovations realized in both time period t and period t-1. The state transition equation for k t+1 has

the form: 1211 lnln −+ ++= tkxtkxtkt xxkk φφφ . The decision rule for a control variable has the form:

121lnln −++= tzxtzxtzkt xxkz φφφ , where zt represents any control variable in the system. To derive

the ARMA structure for the controls, we substitute out the state lnk t in the control equation using the

state transition equation to get

222111211 )()(lnln −−− −+−+++= tzxkkxzktzxkkxzkzxtzxtkt xxxzz φφφφφφφφφφφ . (3.1)

The monetary shock is assumed to follow a first-order autoregressive, AR(1), process as in

Christiano (1991) and Christiano and Eichenbaum (1992):

x xt t t= +−ρ ε1 . (3.2)

Then the equation for zt becomes an ARMA(2,2) given by

1112121 )(lnln)(ln −−− −+++−+= tzxkkxzkzxtzxtktkt zzz εφφφφφεφρφφρ (3.3)

Notice that all of the control variables share the same autoregressive coefficients but different moving

average coefficients, and that the dynamic structure of the exogenous shock (reflected by ρ) affects

only the autoregressive coefficients, not the moving average coefficients.11 Since x is eliminated in

Eq. (3.3), we can avoid measuring the (exogenous) money supply.

For the purpose of our current analysis, we form a VAR for the growth rate of consumption, the

inflation rate, and the output-capital ratio. Appendix C shows that the stochastic processes of these

variables can be expressed in a CVARMA (2,3) model.

3.2. MRS-based and MRT-based Interest Rates

11 If the monetary shock follows a higher order autoregressive process, the ARMA structure in Eq. (3.3) will beaffected. We find, however, that this consideration does not affect our empirical results qualitatively.

.)( 222 −−+ tzxkkxzk εφφφφ

10

The assumption of conditional normality about the fundamentals helps reduce nonlinear

expectations in the theoretical MRS and MRT to the linear ones involving only conditional mean and

variance of the observables. Consider the stochastic process given by

ttiy Ω+ |1, ∼ ))(),(( 1,1, ++ tittit yVyEN , (3.4)

where yit is a time series from the variable set )/ln( ,ln ,ln tttt kqPc ∆∆ , tΩ is the information set

available at time t, and )(⋅tE and variance )(⋅tV are the conditional mean and variance, respectively.

Note that )exp( 1, +tit yE )(exp 1, += tit yE )(5.0 1, ++ tit yV from the conditional lognomality. The

log-linearized equilibrium relationships described in Section 3.1 and Appendix C imply that the

consumption growth, inflation and the logarithm of the output-capital ratio conditional on the

information set are jointly normally distributed.

To calibrate the MRS-based nominal rate that relies on the consumption side, we rewrite Eq.

(2.9) using Eqs. (2.1) and (3.4) as:

ttttMRSt PcER ξγβη +∆−∆−=+ −

++1

11 )]lnlnexp( [1

.)ln,ln()ln(5.0

)ln(5.0)ln()ln(exp)(

111

12

111

tttttt

tttttt

PcCovPV

cVPEcE

ξγγγβη

+∆∆−∆−∆−∆+∆=

+++

+++−

(3.5)

To calibrate the MRT-based nominal rate that relies on the production side, we can express Eq.

(2.11) using Eqs. (2.1) and (3.4) as:

,

)lnlnexp()lnlnlnexp(ln1

11

2121

++

++++

∆−∆−∆−∆−∆−=+

ttt

tttttMRTt PcE

PccNMPKER

γγγβ (3.6)

where )( 1,1 δη −+≡ ++ tkt fNMPK . The calibrated ex-ante real rate is given by:

1 11

11 0 5

11 1+ = +

+= + − +

++ +rr R E R E P V Pt t t

tt t t t t

* ( ) ( ) ( ) exp ( ln ) . ( ln ),π

∆ ∆ (3.7)

where Rt is either RtMRS or Rt

MRT defined by Eqs. (3.5) or (3.6). Hence, using condition

MRS=MRT from Eqs. (3.5)−(3.6) and taking historical means, we have

11

21

12121 lnlnlnexpln

11

∆−∆−∆−

= ∑ =++++

T

tttttt PccNMPKET γγ

ηβ . (3.8)

All variables in Eqs. (3.5)−(3.8) can be replaced by their estimates from the CVARMA model.

Appendix D provides the steps for calibrating the MRS-based and MRT-based interest rate series.

3.3. Risk-Free Rate Puzzle and Keynes-Ramsey Rule

Eqs. (3.5) and (3.6) show that the MRS-based real rate is an increasing function of γ and that

the MRT-based real rate is a decreasing function of γ. This suggests that if we look only at the

consumption-based MRS alone, then a high value of γ needs to be assumed in order to sustain a

higher risk-free rate. On the hand, if we look only at the production-based MRT alone, then a low

value of γ needs to be assumed in order to sustain a higher risk-free rate. However, the steady-

state equilibrium condition, MRS=MRT, implies a unique value for γ. This means that, in a general

equilibrium model, we do not have the freedom of assigning arbitrary value for γ once other

structural parameters (such as β) of the model are specified. The situation is depicted in Fig. 1.

As Mehra and Prescott (1985) suggest, to explain a high equity premium using a conventional

consumption-based approach, a very high value of γ (e.g., above 10) should be assumed. Such a

high value is not plausible empirically, as implied by findings from micro data (see, e.g., Prescott,

1986). This mirrors the risk-free rate puzzle that the return on the default-free asset is too low to be

explained by a consumption-based model (Weil, 1989).12

On the other hand, a production-based approach can resolve the risk-free rate puzzle with only

a low value for γ. Intuitively, the firm, as a member of the family, discounts future returns to a greater

extent if the household is more risk averse. Thus, the less risk averse the household is, the higher the

borrowing costs (future returns) is required by the firm, implying a negative slope of the schedule for

MRT-based real rates. This implies that the risk-free rate puzzle does not exist from the viewpoint of

a firm as a family member.13

12 The existing literature suggests that the risk-free rate puzzle can be resolved by incorporating preferencemodifications (generalized expected utility or habit formation) and incomplete markets in the consumption-basedapproach (see, for the listing of related studies, Kocherlakota 1996).13 The fact that the MRT-based rate is a decreasing function of γ suggests that the risk free rate puzzle may besolved by shedding light on the production side, whereas a consumption-based model should relax standardassumptions through preference modifications or incorporation of incomplete markets to solve the puzzle.

12

Here, we argue that it is theoretically inconsistent to assume an arbitrary value of γ in a general

equilibrium model in order to resolve the risk-free puzzle when we account for both the consumption

and production sides. Namely, the risk aversion parameter is not a free parameter in general

equilibrium.

Specifically, Fig. 2A depicts the relation between β and γ implied by Eq. (3.8), on the basis of

actual data that will be described in Section 4.1. It shows the schedule of β as an increasing function

of γ (the grid for γ is 0.1). Once β is given, then γ is determined. For example, when β = 0.9935,

we have γ = 0.5. Fig. 2B shows how the MRS-based and MRT-based real rates in our model

determine γ, given a specific value of the discount factor, 9935.0=β . The two blades of scissors,

MRS and MRT, determine the value of γ, given β. For an alternative value of β, the two blades

cross at a different location and pick up a different value of γ.

4. Empirical Investigations

4.1. Data

Most time series of the US economy that we use are taken from Citibase for the post-Korean

War period 1954:1−1992:4. We use the rate of return on the three-month Treasury bill as the

nominal interest rate. All the variables except interest rates are seasonally adjusted.

We construct the quarterly series of the marginal productivity of capital by calibrating a Cobb-

Douglas function under the CRS assumption on production technology. The marginal productivity of

capital net of depreciation is measured as:

δα −= )/( ttt kqNMPK , (4.1)

where tq is the real GNP per capita and tk is the net capital stock per capita in the business

sector (see Appendix B).14 Parameter α is set at 0.296, which is calibrated as the capital income

share in the business sector following Cooley and Prescott (1995).15 Also, we estimate that

14 The use of a broad measure of the capital stock that includes the household capital (consumer durables) andgovernment capital provided qualitatively similar results for a variety of tasks performed in this paper.15 We also estimated the production function and capital share equation simultaneously under the restriction ofCRS by the full information maximum likelihood method. This exercise provided a similar estimate for α.

13

=δ 0.049 and =η 1.0036. The sample average of NMPKt is 12 percent per year. The predicted

NMPKt is obtained after replacing the actual with the predicted q/y ratio in Eq. (4.1).16 The ex-ante

real interest rate from the data, rrte, is measured as:

)ln(5.0)ln(exp)1()1

1()1(1 111

+++

∆+∆−⋅+=+

+≡+ tttttt

tte

t PVPERERrrπ

, (4.2)

where Rt is the three-month Treasury bill rate.

To obtain conditional expectation of variables, the CVARMA(2,3) model given by Eq. (c.4) in

Appendix C is estimated by a nonlinear seemingly uncorrelated regression (SUR) method to reflect

that we impose parameter restrictions on the system in which error terms are correlated across

equations. The variable set is: ))'/ln( ,ln ,ln( ttttt kqPcy ∆∆= , where ct is the per capita real

consumption expenditure on nondurables and services and Pt is the consumer price index. The

constraints on the parameters are imposed as implied by the theory. The estimated result of

CVARMA is summarized in Table 1. As shown by 2R , equations for inflation and output-capital

ratio fit the data quite well, whereas the consumption growth equation shows a low value of 0.15.

The restriction implied by Eq. (c.4) is not rejected at the 5 percent level. Although the parameter

restriction can be rather strong as implied by its p-value of 0.04, we impose this restriction to be

consistent with the theory.

4.2. Matching Moments

The mean and standard deviation of the calibrated series based on the MRS and MRT for the

1954:3−1992:4 period are computed by applying the simulation method to Eqs. (3.5)−(3.7). The

ex-ante real rate of interest from the actual data is constructed using the expected inflation computed

from the CVARMA. The mean and standard deviation of ex-ante real rates (per annum) is 1.41

percent and 2.49, respectively. We consider the selected values of the risk aversion

parameter, =γ 0.1, 0.3, 0.5, 0.7, 1.0, 1.5, 2.0, 5.0 and the corresponding time preference

parameter, β, determined by the Keynes-Ramsey rule.

16 The NMPK shows a downward trend. Business cycle models, however, suggest a stationary process withouttrends for the NMPK. We thus linearly detrend the estimated NMPK in calibrating the MRT-based interest rate.

14

Table 2 reports the mean and standard deviation of calibrated real and nominal rates with the

combination of β and γ implied by Eq. (3.8). We obtain notable implications from matching

moments. First, in terms of matching means, there is an upward bias in the MRS-based real rate,

rrMRS, reflecting the risk-free rate puzzle. Second, strong similarities are found between rrMRT and the

actual data. A higher NMPK is required for assuring positive values of rrMRT as consumers are more

risk averse and more impatient. This suggests how the ‘high’ marginal productivity of capital (12

percent) reconciles the ‘low’ risk-free rate. The marginal productivity premium over the risk-free

rate is due to the time to receive cash flow from physical investments as well as the risk aversion in

a growing economy. Third, since the variability of rrMRS sharply rises with a higher γ, matching

second moments enables us to exclude the cases of γ below 0.5 and above 2. Also, too much

variability is involved in all series if 5.1>γ while there is too little variability in RMRT and rrMRS if γ

<0.3.

Finally, we set 5.0=γ and 9935.0=β , with which the mean and standard deviation of interest

rates are explained reasonably well by the calibrated series. Specifically, both real and nominal rates

have an upward bias in the mean by about 0.5 percentage point, much moderating the risk-free rate

puzzle. The upward bias in the MRT-based series arises too, as a result of imposing the Keynes-

Ramsey rule. In terms of variability, rrMRT matches well the actual data whereas rrMRS has a

downward bias of 1.48. RMRT is somewhat less variable while RMRS is a bit more variable than the

actual data. We henceforth use these parameter values in assessing the closeness between the

calibrated series and actual data.

4.3. Prediction

We first take a look at how the calibrated series explains the historical movements of the actual

data. Fig. 3 shows the calibrated series (lines with symbols) along with the actual series for ex-ante

real and nominal rates of interest. Real rates are depicted in Windows A and C. The MRT-based

series (Window A) predicts well the actual data for most of the period. The MRS-based series

(Window C) predicts the actual series quite well before 1980 but does not after 1980, and it is less

volatile than the actual series. Apparently, the drastic rise in the mean of real rates the early 1980s is

not explained by both of calibrated series. Nominal rates are depicted in Windows B and D. Both

15

calibrated series move around the actual data rather closely before 1980. They, however, deviate

much from the actual data in the first half of the 1980s: in particular, the MRS-based series

sometimes violates the non-negativity of the nominal rate due to negative expected inflation rates.

Now we assess the similarity between the actual data and calibrated series in two dimensions.

For this purpose, we use not only the levels of series but also detrended series with different filters

since the levels of interest rates may not be covariance stationary. Namely, we also use the first-

differenced series; series detrended by the Hodrick-Prescott (HP, 1997) filter; and the series

detrended by the band-pass (BP) filter for the 6-32 quarter business cycle frequency as described in

Baxter and King (1995), with the caveat that the use of the HP filter may cause spurious cycles

(e.g., King and Rebelo, 1993).

In the first dimension, we estimate and compare autocorrelation and spectral density functions of

the actual and calibrated series. To save space, we display figures only for the cases with the first

difference and BP filter. Fig. 4 depicts estimated autocorrelation functions for the actual data (solid

lines), MRT-based series (lines with symbols), MRS-based series (dotted lines). For the real rate,

both MRS and MRT show close similarities in the oscillation of the autocorrelation function to the

actual data, with MRT displaying better matches under both the first-difference and BP filters. For

the nominal rate, the MRS and MRT perform reasonably well under the BP filter, but rather poorly

under the first-difference filter. This implies that the model explains the nominal rate better at lower

frequencies because of the Fed’s interest rate smoothing.

Fig. 5 displays estimated spectra. The spectra are normalized by variances, hence the area

underneath each spectral density function is unity. The height of the spectrum at a given frequency

indicates the contribution to the total variance from fluctuations at that frequency.17 For the real rate,

both the MRS and MRT show remarkable predictive power on the data regardless of the filter

used, with the exception that the MRS fails to generate enough fluctuations at the very low frequency

(see Window C). This is consistent with Fig. 4C where the MRS has a weak autocorrelation at lags

longer than 8. For the nominal rate, neither the MRS nor the MRT can explain the dynamics of the

data under the first-difference filter (Window B): they show too much power at the high frequency

17 Cycles per quarter equal to frequency/2π, where the frequency ranges from 0 to π. The period of the cycle is theinverse of cycles per quarter.

16

interval to explain the actual data’s high power at lower frequencies only. This is perhaps because

the model does not account for the Fed’s interest rate smoothing. Under the BP filter (Window D),

on the other hand, the MRS performs very well in explaining the data while the MRT does not.18

This is a reversed situation compared to the case of the real rate, indicating that the productivity-

based approach better explains the real rate movement while the consumption-based approach

better explains the nominal rate movement (at the business cycle frequency).

Table 3 provides summary statistics of the relative mean square approximation errors

(RMSAEs) based on the spectra with different filters. The RMSAEs are suggested by Watson

(1993) as a measure of fit and are similar to 1−R2 in a standard regression: the smaller the better

(see also Yi, 1998). The second panel reports the statistics for each series. First, the MRT-based

real rate achieves a very good fit, regardless of the choice of filter (e.g., the RMSAEs for rrMRT in

column 2 are less than 0.05). The MRS-based real rate also achieves a good fit (as indicated by

RMSAEs for rrMRS in column 4), although, overall, it is not as good as the MRT-based real rate.

Second, the nominal rate is not well explained by the model as the real rate. At the business cycle

frequency, however, the fit is quite good. The MRS-based series has rather lower RMSAEs than

the MRT-based series.

In the second dimension, we assess the similarity of two time series more directly. Table 4

shows the cross-correlation coefficients between the actual and calibrated series. For the real rate,

the MRT-based series outperform the MRS-based series. Regardless of the choice of filter, the

correlation for rrMRT is quite high, ranging between 0.42−0.66. The correlation for rrMRS ranges

between 0.24−0.39. On the other hand, the model tends to predict better the nominal rate than the

real rate when the calibration is based on MRS, whereas the converse is true when it is based on

MRT. Correlations for RMRT and RMRS are reasonably high in levels (0.510 and 0.632, respectively)

and under the BP filter (0.565 and 570, respectively). As shown in Fig. 6, the model largely fits well

the actual data at the business cycle frequency, although the model does not predict smooth interest

18 In levels, we found the astonishing similarity between the spectra of rrMRT and the actual data but lesssimilarity between the spectra of rrMRS and the actual data, and found too much power of the nominal rate at lowfrequencies to be explained by the calibrated series. With the use of the HP filter, the calibrated series showedstrong similarities to the real rate, whereas neither of the two calibrated series captures the high persistence ofthe nominal rate.

17

rates in the 1960s, due to the Fed’s emphasis on financial market stability,19 and a drift in real and

nominal rates in the early 1980s. Remarkably, much of the short-term dynamics of nominal rates is

not explained by our model prediction: under the first-difference and HP filters, correlations for

nominal rates are too low for the model to explain nominal rates.

We now provide implications from the comparison between model predictions and actual data.

First, both sides capture dynamic aspects of the real rate, for which MRT outperforms MRS. MRT

has information superiority to MRS at high frequencies while MRT’s information superiority is much

reduced at low frequencies. This is perhaps because calibration based on MRT utilizes information

on production technology as well as consumption growth whereas MRS does not include the

liquidity effect component, which itself contains information on the production side. Second, the

model predicts well the nominal rate at the business cycle frequency but not at high frequencies. This

finding implies that the nominal rate can be explained by fundamentals of thrift and productivity to

some extent although its fluctuations at high frequencies can be influenced by monetary policy.

4.4. Discussion

Although we have provided a variety of diagnostics for the performance of the model, our

main purpose is not to test the model against a range of other models, but to argue that MRS and

MRT should be jointly examined in a single framework. By doing so, we attempt to reveal

consistency between the implications of MRS and MRT in a general equilibrium model. As argued

by Cochrane (1996, p. 573), there are many factors that can ‘delink’ MRS from interest rates. In

our model, the liquidity effect appears through portfolio rigidity in a monetary production economy.

The unobservable liquidity effect term has been treated as a stochastic disturbance in calibrating the

MRS-based interest rates. The MRT-based interest rates, however, utilize information on both

production technology and preferences, and thus embodies the liquidity effect. As a result, the

wedge between the two series is attributable to the liquidity effect term.20

19 In the 1960s, monetary policy was pursued to maintain the stability of financial markets, focusing on themovements of free reserves to control the bank loan rate or the return on long-term bonds.20 The wedge could also be due to misspecification of the model. The existence of the wedge due to othersources that renders MRS different from the conventional formula implies that a production-based asset pricingmodel gets around the puzzle (see also footnotes 14-15).

18

A friction is imposed on the production side of the economy since firms are assumed to take a

quarter to build the capital stock from investment and another quarter to receive cash flow from

output sales.21 As a result, a compensation for the time to receive cash flow and the risk involved

in physical investment render the high marginal productivity compatible with the low risk-free rate:

note that marginal utility from future consumption is involved in MRT, too. In practice, however,

firms face longer gestation lags (e.g., 2-4 quarters) in installing new capital. The incorporation of the

time-to-build idea (e.g., Kydland and Prescott, 1982) will involve longer period lags in the interest

rate dynamics in conjunction with the inflation dynamics and the term-structure of interest rates.

Our model does not predict an abrupt upward drift in real interest rates in the early 1980s. A

conventional hypothesis is that the real interest rate follows a stationary process with a constant

mean. The time-series literature on real interest rates (e.g., Nelson and Schwert, 1977; Rose, 1988;

Garcia and Perron, 1996), however, has provided evidence against this hypothesis for the post-

WWII period. A simple way to relax the hypothesis is to allow for mean drifts. We employ a

statistical procedure following Quandt (1958) to identify structural break dates assuming the

existence of two breaks. This exercise suggests that two break dates for the ex-ante real rate are

73:2 and 80:3.22 The break at 73:2, plausibly associated with the first oil shock, is captured only by

the MRT-based series. Thus, this break seems to be arising from productivity and inflation factors

embodied in our model. The break at 80:3, however, is explained neither by the MRT-based series

nor by the MRS-based ones.23 We attribute the break in the early 1980s to changes in institutional

factors. The highest and most volatile interest rates in post-WWII history were preceded by the

change in the operating target from the Federal funds rate to nonborrowed reserves in October

1979 and the Depository Institutions Deregulation Act of 1980 that eliminated all deposit interest

21 A typical observation is that firms hold the one-to-three months’ worth of sales (e.g., Bils and Kahn, 1996). Themedian of the mean of the manufacturing firm data from COMPUSTAT (for 1975-1994) shows the yearlyinventory/sales ratio is 0.244 (Choi et al., 1997), implying that firms holds about three months’ worth of sales.22 Break points, (t1, t2), are chosen to minimize the log of the sum of squares residual function: =LSSR

)ˆˆln(*)()ˆˆln(*)()ˆˆln(* 3'322

'2121

'11 eetTeetteet −+−+ , where ie is the sub-period residual of the regression, t

et errrr +=

with et being covariance stationary. Allowing for a Markov switching in the inflation process instead, Garcia andPerron (1996) suggest mean drifts in the ex-ante real rate occurring around the same dates as ours.23 The break dates are 73:1 and 81:4 for the MRT-based series, and 68:4 and 82:3 for the MRS-based series.

19

rate ceilings. Also, fiscal policies and the government debt in the 1980s may have caused high

interest rates.24

More importantly, the nominal rate shows much smoother movements than the calibrated series

in short-time horizons as indicated by the spectra estimates (see also footnote 17) and Windows B

and D of Fig. 4. This finding provides an important policy implication. Our model assumes that the

monetary growth follows an autoregressive process. However, the Fed’s concern with the financial

market stability and interest rate smoothing, indeed, has affected the nominal interest rate

(Rudebusch, 1995; Choi, 1999). Also, inflation targeting would alter the dynamics of interest rates:

e.g., Fuhrer and Moore (1992) suggest that aggressive inflation targeting raises the variability of

interest rates relative to that of inflation. Hence, we expect the Fed’s monetary policy based on a

feedback rule to be able to explain a substantial portion of the discrepancy between the nominal rate

and calibrated series.

5. Conclusion

This paper examines the behavior of interest rates by linking the consumption-based and

production-based approaches into a general equilibrium framework for a growing monetary

economy. We derive two theoretical nonlinear relationships that link interest rates to thrift and to

productivity and calibrate historical time series consistent with the theoretical model using the

constrained VARMA estimation and simulation methods.

We find that the movement of the real rate can be explained to some extent by thrift and quite

well by productivity,25 which provides insight on the risk-free rate puzzle. The calibrated series

based on theoretical relationships, however, fail to explain the abnormal drift in the real rate in the

early 1980s, which is presumably due to institutional factors. We also find some similarities between

the actual and calibrated series for the nominal rate. Nonetheless, the nominal rate shows an

excessive smoothness compared with the calibrated series although these series show close

24 Fitoussi and Phelps (1988) suggest that the US fiscal policy has been accompanied by the high interest ratearound the world from 1980 onwards.25 If the wedge between the MRS and the MRT is indeed due to the liquidity effect, then our approach can alsoshed light on the nature of the liquidity effect itself. Since this issue is quite involved is beyond the scope of thecurrent project, we leave it to future explorations.

20

similarities at the business cycle frequency. It is then puzzling why the calibration of the nominal rate

by the Lucas-Fuerst type general equilibrium model fails to deliver the smooth movement of the

nominal interest rate at high frequencies.

To solve this puzzle, the extension of the model in the following directions, left for future study,

will be helpful. First and most importantly, the incorporation of an endogenous monetary policy that

aims to smooth market rates (and perhaps to target inflation) will contribute to capturing the little

variability of nominal rate movements at high frequencies.26 Second, taking into account the time-to-

build idea and elaborating the production process with the capacity utilization idea (e.g., Bils and

Cho, 1994) may improve the performance of calibrating the MRT-based interest rate. Furthermore,

to understand the puzzle, it may be beneficial to consider the role of government debt (Mankiw,

1987; Evans, 1987) and the sources of friction that may affect the persistence of interest rates such

as incomplete asset markets (Telmer, 1993), adjustment costs (Cochrane, 1991, 1996; Cogley and

Nason, 1995), and transaction costs (Luttmer, 1996).

26 Restrictions on the goods market’s price adjustment speed will affect the dynamics of inflation. Theintroduction of sluggish price adjustments may render movements of expected inflation and thus nominal ratessmoother.

21

Appendix A. Derivation of Eqs. (2.8) and (2.11)

Equilibrium conditions of the model include market clearing conditions for the five markets:).,(=)-(1-+ and , , , ,1 11 tttttttttttttt hkfkkckxdblhmm δηκ ++ =+==== Let 1λ and λ2

denote the Lagrange multipliers associated with (2.5) and (2.6), respectively. The first-orderconditions with respect to indicated variables, evaluated at the equilibrium are:

dt: )]1/()( [)( 1111 ttmtttt xkVREE += +−− βηλ , (a.1)ct: )]1/()( [)(' 11 ttmttt xkVpcU ++= +βηλ , (a.2)lt: )1/()( )1/(1 1 tttmt xwkVl +=−− +βη , (a.3)bt: )1/()( 12 ttmtt xkVR += +βηλ , (a.4)ht: )]1/()( [)1/()( 1211, ttmttttmtht xkVwxkVfp ++=+ +++ βηλβη , (a.5)k t+1: )]1/()( [)( 121 ttmtttk xkVpkV ++= ++ βηλβη , (a.6)

:1tλ ,1 ttt cpd =− (a.7):2tλ ttttttt hwkkpxd +−−=+ + ))(( 1 δη , (a.8)

where ]/)('[)( 111 +++ = ttttm pcUEkV and )].1/()()-( [)( 1211,1 +++++ ++= ttmttkttk xkVpfEkV δηβη

Combining (a.2) and (a.4) yields Eq. (2.8). Combining (a.4) and (a.6) yields Eq. (2.11).

Appendix B. Measuring the marginal productivity of capital

The quarterly series of the capital stock is constructed by using the annual series of capitalstocks (Department of Commerce, 1993; Bureau of Economic Analysis, 1993) and relatedquarterly data. The quarterly data on private fixed investment and the stock of inventories are takenfrom Citibase. The quarterly depreciation rates are generated by using the consumption of fixedcapital stock (GCCJQ in Citibase) and a proper interpolation of annual depreciation data from theDepartment of Commerce (1993). The private capital stock includes the fixed reproducible privatecapital stock and the stock of inventories. The share of capital income in the business sector output,α, is computed following Cooley and Prescott (1995):

Income) Capital Ambiguous()/+Income Capital sUnambiguou( −= GNPDEPα , where GNP is

the nominal GNP, and DEP is the nominal consumption of fixed capital. The sample mean(standard errors) of the capital income share and the depreciation rate for the 54:1−92:4 period arecomputed as α=0.296 (0.012) and δ=0.049 (0.008), respectively. To obtain per capita values,variables are divided by population (PAN17 in Citibase). Then the net marginal productivity ofcapital is given by Eq. (4.1) in the text.

Appendix C. Derivation of the CVARMADenote ∆ = −1 L , where L is the lag operator. The consumption growth is given by

11112121 )(lnln)(ln −−− −−+++∆−∆+=∆ tcxcxkkxckcxtcxtktkt ccc εφφφφφφεφρφφρ

322211222 )()( −− −−+−−−+ tcxkkxcktcxkkxckcxcxkkxck εφφφφεφφφφφφφφφ , (c.1)

and the inflation rate is given by

22

11112121 )(lnln)(ln −−− −−+++∆−∆+=∆ tpxpxkkxpkpxtpxtktkt PPP εφφφφφφεφρφφρ

322211222 )()( −− −−+−−−+ tpxkkxpktpxkkxpkpxpxkkxpk εφφφφεφφφφφφφφφ . (c.2)

To derive the output-capital ratio, we note

2221112121 )()(lnln)(ln −−−− −+−+++−+= tyxkyxzktyxkkxykyxtyxtktkt qqq εφφφφεφφφφφεφρφφρ ,

12121 lnln)(ln −−− ++−+= tkxtkxtktkt kkk εφεφρφφρ .

Hence the output-capital ratio is given by

tkxyxttkttktt kqkqkq εφφρφφρ )()/ln()/ln()()/ln( 112211 −+−+= −−−−

22212112 )()( −− −+−−++ tyxkyxzktkxyxkkxykyx εφφφφεφφφφφφ . (c.3)

Taken together, we have the following CVARMA form:

where σ φ11 1≡ cx , σ φ12 1≡ px , σ φ φ13 1 1≡ −yx kx , )( 111221 cxcxkkxckcx φφφφφφσ −−+≡ ,)( 111222 pxpxkkxpkpx φφφφφφσ −−+= , )( 211223 kxyxkkxykyx φφφφφφσ −−+= ,

)( 1122231 cxkkxckcxcxkkxck φφφφφφφφφσ +−−−≡ , )( 1122232 pxkkxpkpxpxkkxpk φφφφφφφφφσ +−−−≡ ,)( 2233 yxkyxzk φφφφσ −≡ , )( 2241 cxkkxck φφφφσ +−≡ , )( 2242 pxkkxpk φφφφσ +−≡ , and σ43 0≡ . The

model imposes the following restrictions: (a) the autoregressive roots must be identical across thethree equations; (b) the off-diagonal elements and the coefficient σ43 must be zero. No furtherrestrictions are imposed unless the deep parameters of the model are specified. The ARMAstructure in Eq. (c.4) are imposed on the estimation of the conditional moments of the observableswhen calibrating interest rates.

Appendix D. Calibration of interest rate series using the estimated CVARMAConsider a k-dimensional multiple time series with the sample size T, y yT1 , ,L , generated by

a VARMA(p,q) process:

qtqttptptt uuuyAyAy −−−− +++++++= σσµ LL 1111 ,

∆∆

−

∆∆

+=

∆∆

−−

−

−

−−

−

−

)/ln(lnln

100010001

)/ln(lnln

100010001

)()/ln(

lnln

22

2

2

11

1

1

tt

t

t

k

tt

t

t

k

tt

t

t

kq

P

c

kq

P

c

kq

P

c

ρφφρ

1

23

22

21

13

12

11

000000

000000

−

+

+ tt ε

σσ

σε

σσ

σ

,00

0000

000000

3

43

42

41

2

33

32

31

−−

+

+ tt ε

σσ

σε

σσ

σ)4.c(

23

where µ is a ( )k × 1 vector of intercept terms, the Ai s are kk × coefficient matrices, and ut is a k-dimensional white noise. With restrictions as given by Eq. (c.4), the off-diagonal elements of the Ai sare zero. Define

y y y yt t t kt= ( , , , )'1 2 L )1( ×k , Y y y yT= ( , , , )1 2 L (k T× ),B A Ap= ( , , , )µ 1 L )1( +× kpk , Z y yt t t p= − +( , , , ) '1 1L 1)1( ×+kp ,Z Z Z ZT= −( , ,..., )0 1 1 Tkp ×+ )1( , U u u uT= ( , , , )1 2 L ( )k T× .

Then the CVARMA(2,3) model can be rewritten as: Y BZ U= + . The one-step-ahead conditionalforecast is given by 11

ˆ][ ++ = ttt ZByE , where $B is the estimate of vector B.The MRS-based rate, abstracting the unobservable ξt from Eq. (3.5), is computed as:

)'ˆ5.0ˆexp( )(1 11111 WWyWR ut

MRSt Σ+=+ +

−βη , (d.1)

where tt cy ln[∆= , tPln∆ , ]ln ′tNMPK , ]01[1 γ=W , $ ( )y E yt t t+ +=1 1 , and the estimator of

the covariance matrix is defined by TUUu /'ˆˆˆ =Σ with $ $U Y BZ= − . To compute the MRT-basedrate, Eq. (3.6) requires the two-step-ahead forecast given by )|( 2 ttyE Ω+ ),ˆ|( 1,,12 yyyyE ttt L++= ,generated by the estimated model after replacing yt +1 with $yt +1. Let $Ot and $Ut be forecast valuesand errors of ,ln,[ln 21 ++ ∆ tt cNMPK ,ln 1+∆ tc ]'ln 2+∆ tP conditional on Ωt , respectively. Thenumerator in Eq. (3.6) is computed as:

)'ˆˆ5.0ˆexp( 222 W

TUU

WOW t +β , (d.2)

where ]11[2 −−−= γγW . Similarly, the denominator of Eq. (3.6) can be computed.

24

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Table 1Estimated result of CVARMA

Equation Intercept )( kφρ+ )( kρφ− j2σ j3σ j4σ 2R

tcln∆ : j=1

tPln∆ : j=2

)/ln( tt kq : j=3

0.0002**

(0.0001) 0.0007 =

(0.0004)−0.036**

(0.012)

0.670** 0.269 0.465* 0.377* 0.001 0.114 (0.229) (0.221) (0.236) (0.184) (0.080) 0.670** 0.269 0.046 0.284** −0.199* 0.746 (0.229) (0.221) (0.235) (0.098) (0.091) 0.670** 0.269 −0.520** −0.214** − 0.977 (0.229) (0.221) (0.232) (0.084) −

χ2(5) 11.66 (0.040)Notes: The system is estimated by a nonlinear SUR method for the period 1954:3−1992:4 (standarderrors in parentheses). Each regression equation includes an intercept. A chi-square test is performedfor the parameter restriction in the unconstrained VARMA(3,3), following Gallant and Jorgenson(1979). The test statistic follows a χ2(5) distribution (p-value in parenthesis). ** indicates significance atthe 1% level, * indicates significance at the 5% level, and = indicates significance at the 10% level.

Table 2Mean and standard deviation of calibrated interest rates

β

γ

0.9919 0.9927 0.9935 0.9944 0.9956 0.9977 0.9997 1.0122 0.1 0.3 0.5 0.7 1.0 1.5 2.0 5.0

rrMRT

RMRT

2.04 2.04 2.05 2.05 2.06 2.08 2.11 2.54 (2.84) (2.70) (2.63) (2.64) (2.77) (3.27) (4.01) (9.87)

6.36 6.36 6.36 6.37 6.38 6.41 6.44 6.90 (1.24) (1.45) (1.75) (2.10) (2.67) (3.68) (4.73) (11.23)

rrMRS

RMRS

2.02 2.03 2.03 2.03 2.03 2.04 2.04 2.07 (0.20) (0.60) (1.01) (1.41) (2.02) (3.02) (4.03) (10.05)

6.36 6.36 6.36 6.36 6.35 6.35 6.37 6.37 (3.14) (2.97) (2.86) (2.80) (2.82) (3.13) (3.71) (8.99)

Actual data rre: 1.41 R: 5.73 (2.49) (2.93)

Note: Means and standard errors in parentheses are in percentage per annum for the period54:3−92:4.

Table 3Watson’s test results for the fit of calibrated series: RMSAEs

Variable rrMRT RMRT rrMRS RMRS

Level

First difference

HP filter

BP filter

0.010

0.043

0.022

0.011

0.262

0.307

0.445

0.079

0.149

0.035

0.029

0.036

0.053

0.100

0.116

0.011

Notes: Figures are the RMSAEs (relative mean square approximation errors). The HP filterdenotes the Hodrick-Prescott detrending method, and the BP filter denotes the 6−32 quarterband-pass filter (Baxter and King 1995).

Table 4Cross-correlation between actual and calibrated interest rates

Period rrMRT RMRT rrMRS RMRS

Level

First difference

HP filter

BP filter

0.526

0.422

0.537

0.661

0.510

−0.231

0.180

0.565

0.243

0.393

0.315

0.235

0.632

0.146

0.430

0.570

Note: Calibrated rates are based on the assumption of β= 0 99. 35 and γ =0.5.

1

Fig. 1. Interest rates and risk aversion parameter in a growing economy.

MRS

0.9935 MRT

Fig. 2. Tuning the discount factor (β) and risk aversion parameter (γ).

1+rrSupply of loans: rrMRS

(Thrift)

Demand for loans: rrMRT

(Productivity)

γ

Fig. 3. Calibrated real and nominal interest rates.

Fig. 4. Autocorrelation functions for interest rates.

Fig. 5. Spectra for interest rates.

Fig. 6. BP filtered, calibrated real and nominal interest rates